If `cosalpha+cosbeta+cosgamma=0` , then prove that `cos3alpha+cos3beta+cos3gamma=12cosalphacosbetacosgamma`

Explanation

To prove that if \( \cos \alpha + \cos \beta + \cos \gamma = 0 \), then \( \cos 3\alpha + \cos 3\beta + \cos 3\gamma = 12 \cos \alpha \cos \beta \cos \gamma \), we can follow these steps:

### Step 1: Use the identity for \( \cos 3\theta \)
The identity for \( \cos 3\theta \) is given by:
\[ \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \]
Applying this identity, we can express \( \cos 3\alpha \), \( \cos 3\beta \), and \( \cos 3\gamma \):
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